Pricing Derivatives under Multiple Stochastic Factors by Localized Radial Basis Function Methods

TL;DR

This paper introduces two localized Radial Basis Function methods, RBF-PUM and RBF-FD, for efficiently solving high-dimensional PDEs in financial derivative pricing with multiple stochastic factors, demonstrating high accuracy and computational advantages.

Contribution

The paper presents novel localized RBF methods tailored for multi-factor financial PDEs, highlighting their accuracy, sparsity, and parallelization potential, which are improvements over existing approaches.

Findings

Both methods achieve similar convergence orders.

Numerical experiments confirm high accuracy within reasonable computation time.

Parallelization can significantly speed up the RBF-FD method.

Abstract

We propose two localized Radial Basis Function (RBF) methods, the Radial Basis Function Partition of Unity method (RBF-PUM) and the Radial Basis Function generated Finite Differences method (RBF-FD), for solving financial derivative pricing problems arising from market models with multiple stochastic factors. We demonstrate the useful features of the proposed methods, such as high accuracy, sparsity of the differentiation matrices, mesh-free nature and multi-dimensional extendability, and show how to apply these methods for solving time-dependent higher-dimensional PDEs in finance. We test these methods on several problems that incorporate stochastic asset, volatility, and interest rate dynamics by conducting numerical experiments. The results illustrate the capability of both methods to solve the problems to a sufficient accuracy within reasonable time. Both methods exhibit similar orders of convergence, which can be further improved by a more elaborate choice of the method parameters. Finally, we discuss the parallelization potentials of the proposed methods and report the speedup on the example of RBF-FD.